Drawing Diagonals

Yet this is where a 17th – century reader might have struggled a bit, lest we forget that the term ‘symmetry’ had other meanings before 1794, when Adrien-Marie Legendre recovered the moniker to designate an inversion of spatial orientation, whereby the right-handed becomes left-handed.9
We can think of at least one reader of Desargues who must have fully appreciated the awe and power of this perspective method – Leibniz himself, who indeed might have been surprised to find here something akin to his own preoccupations. But let’s face it; this is the stuff of elementary geometry: a few lines and their intersections here and there, some scaling applications, some symmetry. If F is posited as the eye of the observer, the reasoning is precisely reversed. Undoubtedly, Leibniz will have devised the monad from a multiplicity of models, later surveyed by the philosopher Michel Serres.16 Naturally our own conjecture ought to account, upfront, for Leibniz’s vague use of perspective in his philosophic writings at large. This final provision will prove essential when it comes to determining not only the image of a given transversal line T1 located at distance d from the picture, but that of any transversal line in space. First he takes into account the image of a transversal line Tn located n times further than transversal T1 from the plane of the picture, at a distance dn = n • d. In the end, the inclusion of this vertical axis of symmetry brings the number of points available to help us determine the image of a transversal line passing through T1 and T2 to a staggering six. No model is equivalent to the theory it is meant to subtend. Whether the perceived object is located outside in T1, or inside in F0, the internal and external procedures will yield the same image in F1. Critically for Desargues, two of these points, a and a’, are located inside the black box. When it comes to multiple transversal lines, we note that, if G is the eye of the observer, any transversal line Tn located beyond the confines of the box will map internally to either F0 or G0, depending on the parity of integer n (if n is odd, it will be G0; otherwise F0). When one looks at a moving body, suggests Leibniz, God has tuned one’s internal principles of perception in harmony with the movement of the moving body, while
blocking all communications between the two monads governing the body and the moving body. The diagrams reproduced here should provide enough evidence to shore up the conclusion that the four provisions outlined in the case of the transversal line T1 apply equally to lines T 2 and T 3, and by extension, to any line Tn. The whole thing may perhaps be a little more complex than the ‘enchanted description of a palace in a novel’ which, upon reading the author’s draft Brouillon Project Sur les Coniques,% Descartes had admonished Desargues to pursue. Starting with point a’’, located directly opposite point a’ about F0, we can trace a line a’’G intersecting the horizontal F1G1 in point b, from which we can draw another line to point F. To remain ‘within the box’, as it were, we will simply swap vantage points and look for the elevations of the images of any point a between T1 and T2 on the opposite diagonal segment GF0, between h and i2. b lies on a sight line originating in F and ending in a’’’’, a new point located directly opposite a’ about a vertical axis through h1, halfway across rectangle GG0F0F. And as for the swapping of the viewing points F and G, this expedient cannot fail to remind us of Leibniz’s fundamental distinction between perception and apperception. It is on a purely internal basis that the vertical boundary F0G0 determines the reflected image taken from G. The same applies to determining the image of a line T3 located three times further than transversal T1 (n=3) from the plane of the picture. This line meets segment hii2 at the desired elevation, notated /a. Given that ‘it has no windows through which to come and go’,12 each monad is solely determined by itself. It is precisely on such a double regimen of interiority/exteriority that Desargues’s Distance Scale is based. 1 On the basis of this purely internal principle,13 a given monad will be the locus of changing perceptions,14 independent of any other source. Since our goal is to determine from within the elevation of the image of transversal line T1, which happens to coincide with the midpoint of the box’s vertical edge, is it not simpler to look for the intersection of the box’s two diagonals, leaving aside all reflections in F1 or G1? Let us for instance determine the elevation of the image of transversal line T2. It is important to note the key role diagonals play in the general case where a transversal line Tn is located at a random distance from the plane of the picture (this distance no longer being a multiple of d). To figure out the image of a line located two times further than transversal T1 (n=2) from the plane of the picture, Desargues methodically generalises his application.

Updated: 31.10.2014 — 00:12